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This paper is a generalization of a previous analysis of the effects of a small-amplitude, steady, streamwise vorticity field on the flow over an infinitely thin flat plate in an otherwise uniform stream. That analysis, which is given in Goldstein & Leib (1993), required that the disturbance Reynolds number (i.e. the Reynolds number based on the disturbance velocity and length scale) be infinite while the present paper considers the more general case where this quantity can be finite. The results show how an initially linear perturbation of the upstream flow ultimately leads to a small-amplitude but nonlinear cross-flow far downstream from the leading edge. This flow can, under certain conditions, cause the streamwise velocity profiles to develop distinct shear layers in certain localized spanwise regions. These shear layers, which are remarkably similar to the ones that develop in Tollmien–Schlichting-wave transition (Kovasznay, Komoda & Vasudeva 1962), are highly inflectional and can therefore support the rapidly growing inviscid instabilities that are believed to break down into turbulent spots (Greenspan & Benney 1963, and, subsequently, many others). Numerical computations are carried out for input parameters which approximate the flow conditions of some recent experimental studies of the so-called Klebanoff-mode phenomenon. The results are used to explain some of the experimental observations, and, more importantly, to explain why the averaged quantities usually reported in these experiments do not correlate well with the turbulent-spot formation and therefore with the overall transition process.

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This paper is concerned with the downstream evolution of a resonant triad of initially non-interacting linear instability waves in a boundary layer with a weak adverse pressure gradient. The triad consists of a two-dimensional fundamental mode and a pair of equal-amplitude oblique modes that form a subharmonic standing wave in the spanwise direction. The growth rates are small and there is a well-defined common critical layer for these waves. As in Goldstein & Lee (1992), the wave interaction takes place entirely within this critical layer and is initially of the parametric-resonance type. This enhances the spatial growth rate of the subharmonic but does not affect that of the fundamental. However, in contrast to Goldstein & Lee (1992), the initial subharmonic amplitude is assumed to be small enough so that the fundamental can become nonlinear within its own critical layer before it is affected by the subharmonic. The subharmonic evolution is then dominated by the parametric-resonance effects and occurs on a much shorter streamwise scale than that of the fundamental. The subharmonic amplitude continues to increase during this parametric-resonance stage – even as the growth rate of the fundamental approaches zero – and the subharmonic eventually becomes large enough to influence the fundamental which causes both waves to evolve on the same shorter streamwise scale.

This paper is concerned with the effect of a weak spanwise-variable mean-flow distortion on the growth of oblique instability waves in a Blasius boundary layer. The streamwise component of the distortion velocity initially grows linearly with increasing streamwise distance, reaches a maximum, and eventually decays through the action of viscosity. This decay occurs slowly and allows the distortion to destabilize the Blasius flow over a relatively large streamwise region. It is shown that even relatively weak distortions can cause certain oblique Rayleigh instability waves to grow much faster than the usual two-dimensional Tollmien–Schlichting waves that would be the dominant instability modes in the absence of the distortion. The oblique instability waves can then become large enough to interact nonlinearly within a common critical layer. It is shown that the common amplitude of the interacting oblique waves is governed by the amplitude evolution equation derived in Goldstein & Choi (1989). The implications of these results for Klebanoff-type transition are discussed.

We consider the effects of strong critical-layer nonlinearity on the spatial evolution of an initially linear ‘acoustic mode’ instability wave on a hypersonic flat-plate boundary layer. Our analysis shows that nonlinearity, which is initially confined to a thin critical layer, first becomes important when the amplitude of the pressure fluctuations becomes O(1/M4InM2), where M is the free-stream Mach number. The flow outside the critical layer is still determined by linear dynamics and therefore takes the form of a linear instability wave — but with its amplitude completely determined by the flow within the critical layer. The latter flow is determined by a coupled set of nonlinear equations, which we had to solve numerically.

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